L(s) = 1 | − 3.04·2-s − 81·3-s − 502.·4-s − 901.·5-s + 246.·6-s + 1.31e3·7-s + 3.09e3·8-s + 6.56e3·9-s + 2.74e3·10-s + 5.85e4·11-s + 4.07e4·12-s − 1.65e4·13-s − 4.00e3·14-s + 7.30e4·15-s + 2.47e5·16-s − 3.14e5·17-s − 1.99e4·18-s − 5.30e5·19-s + 4.53e5·20-s − 1.06e5·21-s − 1.78e5·22-s − 1.77e6·23-s − 2.50e5·24-s − 1.13e6·25-s + 5.05e4·26-s − 5.31e5·27-s − 6.61e5·28-s + ⋯ |
L(s) = 1 | − 0.134·2-s − 0.577·3-s − 0.981·4-s − 0.645·5-s + 0.0777·6-s + 0.207·7-s + 0.266·8-s + 0.333·9-s + 0.0869·10-s + 1.20·11-s + 0.566·12-s − 0.161·13-s − 0.0278·14-s + 0.372·15-s + 0.945·16-s − 0.912·17-s − 0.0448·18-s − 0.934·19-s + 0.633·20-s − 0.119·21-s − 0.162·22-s − 1.32·23-s − 0.154·24-s − 0.583·25-s + 0.0216·26-s − 0.192·27-s − 0.203·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.5408903882\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5408903882\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 81T \) |
| 59 | \( 1 - 1.21e7T \) |
good | 2 | \( 1 + 3.04T + 512T^{2} \) |
| 5 | \( 1 + 901.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.31e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.85e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.65e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.14e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 5.30e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.77e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 9.69e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.85e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.58e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.46e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 8.74e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 9.62e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.35e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 2.10e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 7.42e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.60e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.75e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.38e6T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.20e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.11e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.55e9T + 7.60e17T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.08727603083129971701136856352, −9.976578512926654353091140555476, −8.974688283955229456806442836349, −8.104743930530570539394702268622, −6.87450019149431781700416132793, −5.70381003414145549172531004182, −4.36994440540912627597193266964, −3.87794390727276562642223574001, −1.76570097931885888712131605744, −0.38658179695341850331658824815,
0.38658179695341850331658824815, 1.76570097931885888712131605744, 3.87794390727276562642223574001, 4.36994440540912627597193266964, 5.70381003414145549172531004182, 6.87450019149431781700416132793, 8.104743930530570539394702268622, 8.974688283955229456806442836349, 9.976578512926654353091140555476, 11.08727603083129971701136856352